A Divergence-Free and H(div)H(div)-Conforming Embedded-Hybridized DG Method for the Incompressible Resistive MHD equations

We proposed a divergence-free and $H(div)$-conforming embedded-hybridized discontinuous Galerkin (E-HDG) method for solving stationary incompressible viso-resistive magnetohydrodynamic (MHD) equations. In particular, the E-HDG method is computationally far more advantageous over the hybridized discontinuous Galerkin (HDG) counterpart in general. The benefit is even significant in the three-dimensional/high-order/fine mesh scenario. On a simplicial mesh, our method with a specific choice of the approximation spaces is proved to be well-posed for the linear case. Additionally, the velocity and magnetic fields are divergence-free and $H(div)$-conforming for both linear and nonlinear cases. Moreover, the results of well-posedness analysis, divergence-free property, and $H(div)$-conformity can be directly applied to the HDG version of the proposed approach. The HDG or E-HDG method for the linearized MHD equations can be incorporated into the fixed point Picard iteration to solve the nonlinear MHD equations in an iterative manner. We examine the accuracy and convergence of our E-HDG method for both linear and nonlinear cases through various numerical experiments including two- and three-dimensional problems with smooth and singular solutions. For smooth problems, the results indicate that convergence rates in the $L^2$ norm for the velocity and magnetic fields are optimal in the regime of low Reynolds number and magnetic Reynolds number. Furthermore, the divergence error is machine zero for both smooth and singular problems. Finally, we numerically demonstrated that our proposed method is pressure-robust

Hybridized discontinuous Galerkin methods for magnetohydrodynamics

Discontinuous Galerkin (DG) methods combine the advantages of classical finite element and finite volume methods. Like finite volume methods, through the use of discontinuous spaces in the discrete functional setting, we automatically have local conservation, an essential property for a numerical method to behave well when applied to hyperbolic conservation laws. Like classical finite element methods, DG methods allow for higher order approximations with compact stencils. For time-dependent problems with implicit time stepping and for steady-state problems, DG methods give a larger globally coupled linear system than continuous Galerkin methods (especially for three dimensional problems and low polynomial orders). The primary motivation of the hybridized (or hybridizable) discontinuous Galerkin (HDG) methods is to reduce the number of globally coupled unknowns in DG methods when implicit time stepping or direct-to-steady-state solutions are desired. This is accomplished by the introduction of new “trace unknowns” defined on the mesh skeleton, the definition of one-sided numerical fluxes, and the enforcement of local conservation. This results in a globally coupled linear system where the local “volume unknowns” can be eliminated in a Schur complement procedure, resulting in a reduced globally coupled system in terms of only the trace unknowns. Magnetohydrodynamics (MHD) is the study of the flow of electrically conducting fluids under the influence of magnetic fields. The MHD equations are used to describe important physical phenomena including laboratory plasmas (plasma confinement in fusion energy devices), astrophysical plasmas (solar coronas, planetary magnetospheres) and liquid metal flows (metallurgy processes, the Earth’s molten core, cooling for nuclear reactors). Incompressible MHD, which is the main focus of this work, is relevant in low Lundquist number liquid metals, in high Lundquist number, large guide field fusion plasmas, and in low Mach number compressible flows. The equations of MHD are highly nonlinear, and are characterized by physical phenomena spanning wide ranges of length and time scales. For numerical methods, this presents challenges in both spatial and temporal discretization. In terms of temporal discretization, fully implicit numerical methods are attractive in their robustness; they allow for stable, high-order time integration over long time scales of interest.Computational Science, Engineering, and Mathematic

Fast and scalable solvers for high-order hybridized discontinuous Galerkin methods with applications to fluid dynamics and magnetohydrodynamics

The hybridized discontinuous Galerkin methods (HDG) introduced a decade ago is a promising candidate for high-order spatial discretization combined with implicit/implicit-explicit time stepping. Roughly speaking, HDG methods combines the advantages of both discontinuous Galerkin (DG) methods and hybridized methods. In particular, it enjoys the benefits of equal order spaces, upwinding and ability to handle large gradients of DG methods as well as the smaller globally coupled linear system, adaptivity, and multinumeric capabilities of hybridized methods. However, the main bottleneck in HDG methods, limiting its use to small to moderate sized problems, is the lack of scalable linear solvers. In this thesis we develop fast and scalable solvers for HDG methods consisting of domain decomposition, multigrid and multilevel solvers/preconditioners with an ultimate focus on simulating large scale problems in fluid dynamics and magnetohydrodynamics (MHD). First, we propose a domain decomposition based solver namely iterative HDG for partial differential equations (PDEs). It is a fixed point iterative scheme, with each iteration consisting only of element-by-element and face-by-face embarrassingly parallel solves. Using energy analysis we prove the convergence of the schemes for scalar and system of hyperbolic PDEs and verify the results numerically. We then propose a novel geometric multigrid approach for HDG methods based on fine scale Dirichlet-to-Neumann maps. The algorithm combines the robustness of algebraic multigrid methods due to operator dependent intergrid transfer operators and at the same time has fixed coarse grid construction costs due to its geometric nature. For diffusion dominated PDEs such as the Poisson and the Stokes equations the algorithm gives almost perfect hp--scalability. Next, we propose a multilevel algorithm by combining the concepts of nested dissection, a fill-in reducing ordering strategy, variational structure and high-order properties of HDG, and domain decomposition. Thanks to its root in direct solver strategy the performance of the solver is almost independent of the nature of the PDEs and mostly depends on the smoothness of the solution. We demonstrate this numerically with several prototypical PDEs. Finally, we propose a block preconditioning strategy for HDG applied to incompressible visco-resistive MHD. We use a least squares commutator approximation for the inverse of the Schur complement and algebraic multigrid or the multilevel preconditioner for the approximate inverse of the nodal block. With several 2D and 3D transient examples we demonstrate the robustness and parallel scalability of the block preconditionerAerospace Engineerin

HDGlab: An Open-Source Implementation of the Hybridisable Discontinuous Galerkin Method in MATLAB

This paper presents HDGlab, an open source MATLAB implementation of the hybridisable discontinuous Galerkin (HDG) method. The main goal is to provide a detailed description of both the HDG method for elliptic problems and its implementation available in HDGlab. Ultimately, this is expected to make this relatively new advanced discretisation method more accessible to the computational engineering community. HDGlab presents some features not available in other implementations of the HDG method that can be found in the free domain. First, it implements high-order polynomial shape functions up to degree nine, with both equally-spaced and Fekete nodal distributions. Second, it supports curved isoparametric simplicial elements in two and three dimensions. Third, it supports non-uniform degree polynomial approximations and it provides a flexible structure to devise degree adaptivity strategies. Finally, an interface with the open-source high-order mesh generator Gmsh is provided to facilitate its application to practical engineering problems

Computational Engineering

The focus of this Computational Engineering Workshop was on the mathematical foundation of state-of-the-art and emerging finite element methods in engineering analysis. The 52 participants included mathematicians and engineers with shared interest on discontinuous Galerkin or Petrov-Galerkin methods and other generalized nonconforming or mixed finite element methods

Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018

This open access book features a selection of high-quality papers from the presentations at the International Conference on Spectral and High-Order Methods 2018, offering an overview of the depth and breadth of the activities within this important research area. The carefully reviewed papers provide a snapshot of the state of the art, while the extensive bibliography helps initiate new research directions